R Notebook

library(GGally)

Registered S3 method overwritten by 'GGally':
  method from   
  +.gg   ggplot2

prestige %>% 
  skim()

── Data Summary ────────────────────────
                           Values    
Name                       Piped data
Number of rows             102       
Number of columns          6         
_______________________              
Column type frequency:               
  factor                   1         
  numeric                  5         
________________________             
Group variables            None      

Drop NAs and remove ‘census’ variable

Feature engineering / variable transformation

• Logged some positively skewed numerics
  • women
  • income

prestige_log <- prestige %>% 
  mutate(ln_women = log(0.1 + women),
         ln_income = log(income))

skim(prestige_log)

── Data Summary ────────────────────────
                           Values      
Name                       prestige_log
Number of rows             98          
Number of columns          7           
_______________________                
Column type frequency:                 
  factor                   1           
  numeric                  6           
________________________               
Group variables            None        

prestige_log %>% 
  select(prestige, everything()) %>% 
  ggpairs(aes(colour = type,
              alpha = 0.5))

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• By looking at the logs we are accounting for the large outliers
• This makes it look more like a normal distribution
• Correlation also increases by 0.05
• The scatter plot of ln_income is much more ‘normal’ than just income

Investigate the large outliers for income:

prestige %>% 
  slice_max(income, n = 10)

Start building our model:

Best predictor first

mod1a <- lm(prestige ~ education,
            data = prestige)

summary(mod1a)

Call:
lm(formula = prestige ~ education, data = prestige)

Residuals:
    Min      1Q  Median      3Q     Max 
-21.605  -6.151   0.366   6.565  17.540 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -10.8409     3.5285  -3.072  0.00276 ** 
education     5.3884     0.3168  17.006  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.578 on 96 degrees of freedom
Multiple R-squared:  0.7508,    Adjusted R-squared:  0.7482 
F-statistic: 289.2 on 1 and 96 DF,  p-value: < 2.2e-16

For every unit increase in education, prestige increases 5.3 if all other variables are held constant. Education can explain 75% of variation in prestige

autoplot(mod1a)

All plots look good - we could accept these

mod1b <- lm(prestige ~ type,
            data = prestige)

summary(mod1b)

Call:
lm(formula = prestige ~ type, data = prestige)

Residuals:
     Min       1Q   Median       3Q      Max 
-18.2273  -7.1773  -0.0854   6.1174  25.2565 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   35.527      1.432  24.810  < 2e-16 ***
typeprof      32.321      2.227  14.511  < 2e-16 ***
typewc         6.716      2.444   2.748  0.00718 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9.499 on 95 degrees of freedom
Multiple R-squared:  0.6976,    Adjusted R-squared:  0.6913 
F-statistic: 109.6 on 2 and 95 DF,  p-value: < 2.2e-16

• Blue-collar: 35.5
• White-collar: 42.2
• Professional: 74.5
• RSE slightly higher (9.5)
• \(r^2\) slightly lower (0.7)

autoplot(mod1b)

Although both models look good, mod1a is better (based on r^2 and RSE), so we would choose ‘education’ as our first predictor

now we want to see which variables describe the ‘residue’ - the unexplained model error

change the variable of interest

prestige_resid <- prestige_log %>% 
  add_residuals(mod1a) %>% 
  select(-c(prestige, education))

prestige_resid %>% 
  select(resid, everything()) %>% 
  ggpairs(aes(colour = type,
              alpha = 0.5))

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things to model next based on this:

• ln_income
• income
• type

add second predictor - the one that explains the most residual error

mod2a<- lm(prestige ~ education + ln_income,
           data = prestige_log)

summary(mod2a)

Call:
lm(formula = prestige ~ education + ln_income, data = prestige_log)

Residuals:
     Min       1Q   Median       3Q      Max 
-16.6775  -4.0506  -0.2538   4.0648  16.7992 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -101.1882    12.8490  -7.875 5.51e-12 ***
education      4.0375     0.3173  12.726  < 2e-16 ***
ln_income     12.0564     1.6719   7.211 1.33e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.932 on 95 degrees of freedom
Multiple R-squared:  0.8389,    Adjusted R-squared:  0.8356 
F-statistic: 247.4 on 2 and 95 DF,  p-value: < 2.2e-16

autoplot(mod2a)

• \(r^2\) is up to 83.4
• RSE is down to 6.9
• education coefficient is reduced to 4 (from 5.3)
• can’t really compare these coefficients as education is measured in years and ln_income is a log of dollars

mod2b<- lm(prestige ~ education + income,
           data = prestige_log)

summary(mod2b)

Call:
lm(formula = prestige ~ education + income, data = prestige_log)

Residuals:
     Min       1Q   Median       3Q      Max 
-16.9367  -4.8881   0.0116   4.9690  15.9280 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -7.6210352  3.1162309  -2.446   0.0163 *  
education    4.2921076  0.3360645  12.772  < 2e-16 ***
income       0.0012415  0.0002185   5.682 1.45e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.45 on 95 degrees of freedom
Multiple R-squared:  0.814, Adjusted R-squared:  0.8101 
F-statistic: 207.9 on 2 and 95 DF,  p-value: < 2.2e-16

autoplot(mod2b)

• less predictive of a model
• education coefficent less effected
• \(r^2\) less
• RES higher

We can reject this model in favour of mod2a

mod2c<- lm(prestige ~ education + type,
           data = prestige_log)

summary(mod2c)

Call:
lm(formula = prestige ~ education + type, data = prestige_log)

Residuals:
    Min      1Q  Median      3Q     Max 
-19.410  -5.508   1.360   5.694  17.171 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -2.6982     5.7361  -0.470   0.6392    
education     4.5728     0.6716   6.809 9.16e-10 ***
typeprof      6.1424     4.2590   1.442   0.1526    
typewc       -5.4585     2.6907  -2.029   0.0453 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.814 on 94 degrees of freedom
Multiple R-squared:  0.7975,    Adjusted R-squared:  0.791 
F-statistic: 123.4 on 3 and 94 DF,  p-value: < 2.2e-16

autoplot(mod2c)

• type labels explain less when education is controlled
• \(r^2\) is down
• RSE is higher

when ‘type’ categories have different levels of significance, either use all or none of them - DON”T CHERRY PICK

We can reject this model in favour of mod2a

Results:

• mod2a gave biggest uplift in \(r^2\)
• residuals look great

check for significance of categorical: ANOVA test

anova(mod1a, mod2c)

Analysis of Variance Table

Model 1: prestige ~ education
Model 2: prestige ~ education + type
  Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
1     96 7064.4                                  
2     94 5740.0  2    1324.4 10.844 5.787e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

the means of the categorical levels are statistically different, therefore all 3 levels could be included

third predictor

prestige_resid <- prestige_log %>% 
  add_residuals(mod2a) %>% 
  select(-c(prestige, education, ln_income))

prestige_resid %>% 
  select(resid, everything()) %>%
  ggpairs(aes(colour = type,
              alpha = 0.5))

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things to model next based on this:

• type
• women

mod3a <- lm(prestige ~ education + ln_income + type,
           data = prestige_log)

summary(mod3a)

Call:
lm(formula = prestige ~ education + ln_income + type, data = prestige_log)

Residuals:
    Min      1Q  Median      3Q     Max 
-13.511  -3.746   1.011   4.356  18.438 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -81.2019    13.7431  -5.909 5.63e-08 ***
education     3.2845     0.6081   5.401 5.06e-07 ***
ln_income    10.4875     1.7167   6.109 2.31e-08 ***
typeprof      6.7509     3.6185   1.866   0.0652 .  
typewc       -1.4394     2.3780  -0.605   0.5465    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.637 on 93 degrees of freedom
Multiple R-squared:  0.8555,    Adjusted R-squared:  0.8493 
F-statistic: 137.6 on 4 and 93 DF,  p-value: < 2.2e-16

autoplot(mod3a)

do an anova test on types

anova(mod2a, mod3a)

Analysis of Variance Table

Model 1: prestige ~ education + ln_income
Model 2: prestige ~ education + ln_income + type
  Res.Df    RSS Df Sum of Sq      F   Pr(>F)   
1     95 4565.4                                
2     93 4096.3  2    469.07 5.3247 0.006465 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

the results are significant - we can include

add type to model

mod3b <- lm(prestige ~ education + ln_income + women,
           data = prestige_log)

summary(mod3b)

Call:
lm(formula = prestige ~ education + ln_income + women, data = prestige_log)

Residuals:
     Min       1Q   Median       3Q      Max 
-16.8202  -4.7019   0.0696   4.2245  17.6833 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -129.16790   18.95716  -6.814 8.97e-10 ***
education      3.59404    0.38431   9.352 4.39e-15 ***
ln_income     15.60547    2.43246   6.416 5.62e-09 ***
women          0.06481    0.03270   1.982   0.0504 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.828 on 94 degrees of freedom
Multiple R-squared:  0.8454,    Adjusted R-squared:  0.8405 
F-statistic: 171.4 on 3 and 94 DF,  p-value: < 2.2e-16

autoplot(mod3b)

Interactions (between main effects)

prestige_resid <- prestige_log %>% 
  add_residuals(mod3a) %>% 
  select(-prestige)

coplot(resid ~ ln_income | education,
       panel = function(x, y, ...) {
         points(x, y)
         abline(lm(y ~ x), col = "blue")
       },
       data = prestige_resid,
       rows = 1)

• over predicting at higher income
• under predicting at lower income

prestige_resid %>% 
  ggplot(aes(x = education,
             y = resid,
             colour = type)) +
  geom_point()+
  geom_smooth(method = "lm", se = FALSE)

`geom_smooth()` using formula 'y ~ x'

• there is a difference in type levels
• model could benefit from including this

prestige_resid %>% 
  ggplot(aes(x = ln_income,
             y = resid,
             colour = type)) +
  geom_point()+
  geom_smooth(method = "lm", se = FALSE)

`geom_smooth()` using formula 'y ~ x'

• there is a difference in type levels
• model could benefit from including this

Including the interactions

mod4a <- lm(prestige ~ education + ln_income + type + education:ln_income,
            data = prestige_log)

summary(mod4a)

Call:
lm(formula = prestige ~ education + ln_income + type + education:ln_income, 
    data = prestige_log)

Residuals:
     Min       1Q   Median       3Q      Max 
-13.6129  -4.1461   0.7695   4.1546  17.9453 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)         -166.7630    51.0506  -3.267 0.001530 ** 
education             11.1444     4.5601   2.444 0.016438 *  
ln_income             20.2741     5.8790   3.449 0.000852 ***
typeprof               6.8626     3.5803   1.917 0.058374 .  
typewc                -2.3516     2.4103  -0.976 0.331796    
education:ln_income   -0.8892     0.5114  -1.739 0.085414 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.566 on 92 degrees of freedom
Multiple R-squared:  0.8601,    Adjusted R-squared:  0.8525 
F-statistic: 113.1 on 5 and 92 DF,  p-value: < 2.2e-16

• education:ln_income is only a ‘.’ significance
  • maybe don’t include

mod4b <- lm(prestige ~ education + ln_income + type + education:type,
            data = prestige_log)

summary(mod4b)

Call:
lm(formula = prestige ~ education + ln_income + type + education:type, 
    data = prestige_log)

Residuals:
    Min      1Q  Median      3Q     Max 
-13.962  -3.797   1.041   4.092  16.503 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)        -81.6672    14.3681  -5.684 1.57e-07 ***
education            3.1407     0.9004   3.488 0.000751 ***
ln_income           10.6833     1.7108   6.245 1.33e-08 ***
typeprof            15.6176    14.2168   1.099 0.274871    
typewc             -30.4466    18.3465  -1.660 0.100451    
education:typeprof  -0.5801     1.2211  -0.475 0.635887    
education:typewc     2.6675     1.7551   1.520 0.132018    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.585 on 91 degrees of freedom
Multiple R-squared:  0.8608,    Adjusted R-squared:  0.8516 
F-statistic: 93.79 on 6 and 91 DF,  p-value: < 2.2e-16

mod4c <- lm(prestige ~ education + ln_income + type + type:ln_income,
            data = prestige_log)

summary(mod4c)

Call:
lm(formula = prestige ~ education + ln_income + type + type:ln_income, 
    data = prestige_log)

Residuals:
    Min      1Q  Median      3Q     Max 
-13.484  -4.453   1.122   4.123  18.737 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        -118.4325    20.3728  -5.813 8.97e-08 ***
education             3.2107     0.5993   5.357 6.31e-07 ***
ln_income            14.9336     2.4928   5.991 4.12e-08 ***
typeprof             82.7757    31.5059   2.627   0.0101 *  
typewc               51.3717    36.8521   1.394   0.1667    
ln_income:typeprof   -8.5690     3.5251  -2.431   0.0170 *  
ln_income:typewc     -6.1925     4.3172  -1.434   0.1549    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.491 on 91 degrees of freedom
Multiple R-squared:  0.8647,    Adjusted R-squared:  0.8558 
F-statistic: 96.96 on 6 and 91 DF,  p-value: < 2.2e-16

autoplot(mod4c)

anova(mod3a, mod4c)

Analysis of Variance Table

Model 1: prestige ~ education + ln_income + type
Model 2: prestige ~ education + ln_income + type + type:ln_income
  Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
1     93 4096.3                              
2     91 3834.2  2    262.13 3.1107 0.04934 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Relative Importance

calc.relimp(mod3a, type = "lmg", rela = TRUE)

Response variable: prestige 
Total response variance: 292.2358 
Analysis based on 98 observations 

4 Regressors: 
Some regressors combined in groups: 
        Group  type : typeprof typewc 

 Relative importance of 3 (groups of) regressors assessed: 
 type education ln_income 
 
Proportion of variance explained by model: 85.55%
Metrics are normalized to sum to 100% (rela=TRUE). 

Relative importance metrics: 

                lmg
type      0.3352553
education 0.3831467
ln_income 0.2815980

Average coefficients for different model sizes: 

             1group   2groups   3groups
education  5.388408  4.305159  3.284486
ln_income 24.619472 12.879804 10.487471
typeprof  32.321114 14.810837  6.750887
typewc     6.716206  1.013706 -1.439403

calc.relimp(mod4c, type = "lmg", rela = TRUE)

Response variable: prestige 
Total response variance: 292.2358 
Analysis based on 98 observations 

6 Regressors: 
Some regressors combined in groups: 
        Group  type : typeprof typewc 
        Group  ln_income:type : ln_income:typeprof ln_income:typewc 

 Relative importance of 4 (groups of) regressors assessed: 
 type ln_income:type education ln_income 
 
Proportion of variance explained by model: 86.47%
Metrics are normalized to sum to 100% (rela=TRUE). 

Relative importance metrics: 

                      lmg
type           0.38512449
ln_income:type 0.01146495
education      0.29662098
ln_income      0.30678958

Average coefficients for different model sizes: 

                      1group   2groups   3groups   4groups
education           5.388408  4.305159  3.284486  3.210707
ln_income          24.619472 12.879804 14.634783 14.933637
typeprof           32.321114 14.810837 54.690682 82.775730
typewc              6.716206  1.013706 41.131305 51.371716
ln_income:typeprof       NaN       NaN -9.001091 -8.568986
ln_income:typewc         NaN       NaN -8.979324 -6.192503